Prove by Mathematical Induction 14| 3^(4n+2) +5^(2n+1)

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Prove that `14|(3^(4n+2)+5^(2n+1))` :

Base step: This is true for n=0 since 14|(9+5); it is also true for n=1 since 14|(729+125)

Assume that for some `k in NN` that `14|(3^(4k+2)+5^(2k+1))`

Show that `14|(3^(4(k+1)+2)+5^(2(k+1)+1))` or `14|(3^(4k+6)+5^(2k+3))` :


Now `14|(3^(4k+2)+5^(2k+1))==> 3^(4k+2)+5^(2k+1)-=0("mod"14)`

Then `(3^(4k+2)("mod"14)+5^(2k+1)("mod"14))("mod"14)-=0("mod"14)` ``So `3^(4k+2)("mod"14)-=-5^(2k+1)("mod"14)`







`==>14|(81*3^(4k+2)+25*5^(2k+1))` as required.

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